Sum-product number¶

Sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory.

Version: v1 · 2026-09-28 · History
Domain-specific #: 12364
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Recreational Number Theory → Mathematics

Core Idea¶

A sum-product number in base b is a natural number equal to the product of the sum of its base-b digits and the product of those digits. If S_b(n) is the digit sum and P_b(n) the digit product, the defining fixed-point equation is n = S_b(n)P_b(n). The property is representation-dependent: changing the base changes the digits, the function, and generally the membership of n. In decimal notation, 144 is a nontrivial example because (1+4+4)(1×4×4)=9×16=144.

Viewing F_b(n)=S_b(n)P_b(n) as a map on the natural numbers places the concept in arithmetic dynamics. Sum-product numbers are its fixed points; periodic orbits of longer length give sociable sum-product numbers, and iterating from an arbitrary seed produces a transient followed by a cycle. Digit bounds imply that for sufficiently many digits F_b(n) is smaller than n: the largest possible sum grows linearly with digit count and the largest product exponentially with base-dependent factor (b−1), while the smallest number with that many digits grows as b to a power. Consequently only finitely many fixed points exist in each base and all orbits eventually enter a bounded region.

The defining equality also implies divisibility by the digit sum, so every sum-product number is a Harshad number in the same base, but the converse need not hold. Any numeral containing a zero digit has digit product zero and therefore cannot be a positive nonzero fixed point. The abstraction is not an intrinsic magnitude class like primality; it is a base-sensitive fixed-point relation between a number and two aggregate functions of its digit representation.

Structural Signature¶

Sig role-phrases:

the numeral base — base $b$ fixing the digit alphabet and positional representation
the represented integer — natural number decomposed into its base-$b$ digits
the digit-sum statistic — additive aggregate $S_b(n)$ of all digits
the digit-product statistic — multiplicative aggregate $P_b(n)$, forced to zero by any zero digit
the sum–product map — transformation $F_b(n)=S_b(n)P_b(n)$
the fixed-point equation — equality of the original integer and the map output
the Harshad consequence — divisibility by the digit sum for every positive fixed point
the base-dependence rule — membership changing when representation changes
the finite-state bound — positional growth eventually exceeding possible digit aggregates, forcing all orbits into a bounded region
the dynamical extension — transients and longer periodic cycles surrounding the period-one sum-product numbers

What It Is Not¶

Not an intrinsic magnitude class like primality. Membership depends on the numeral base and its digits rather than on arithmetic structure alone.
Not any number divisible by its digit sum. Every sum-product fixed point is Harshad, but divisibility alone does not force equality with sum times product.
Not possible for a positive numeral containing zero. Its digit product vanishes, so the defining right-hand side cannot equal a positive n.
Not a longer periodic orbit. Perfect sum-product numbers are period-one fixed points; sociable variants return only after several iterations.
Not invariant under changing notation. The same integer can satisfy the equation in one base and fail it in another.
Not an infinite family at a fixed base. Digit bounds force sufficiently long numbers downward, confining fixed points to a finite region.
Not merely sum of digits or product of digits separately. The defining map multiplies both aggregates and compares the result with the original number.

Scope of Application¶

Sum-product number applies to base-dependent fixed points of the digit map n equals digit sum times digit product, and to the finite arithmetic dynamics generated by iterating that map.

Fixed-point enumeration. Candidates are generated and checked exactly under one base and numeral convention.
Finite-state dynamics. Growth bounds force trajectories into a bounded region where cycles and basins can be exhaustively studied.
Sociable cycles. Longer periodic orbits extend the fixed-point question under the same map.
Harshad relations. Divisibility by the digit sum is necessary for positive fixed points but not sufficient.
Base comparison. The same integer can qualify in one base and fail in another because its digits change.
Completeness proofs. Place-value growth versus maximum digit sum-product establishes a finite search region.
Algorithm verification. Domain, leading-zero rule, search bound, and cycle detection support reproducible computation.
Applicability boundary. This is not primality or another intrinsic number class, zeros prevent positive fixed points through a zero product, leading zeros cannot be admitted, and computation without a proved bound does not establish completeness.

Clarity¶

Sum-product number denotes a fixed point of the base-dependent map that multiplies a number's digit sum by its digit product. It is distinct from numbers equal only to a digit sum or product and from longer periodic orbits under iteration. Because zero digits force the product to zero and changing base changes every digit, the base and treatment of zero are structural, not incidental. The sharper arithmetic question is how digit bounds restrict possible fixed points and which candidates satisfy $n=S_b(n)P_b(n)$ exactly.

Manages Complexity¶

Sum-product numbers compress a search over integers into the digit sum, digit product, base, and a fixed-point equation. Zero digits eliminate most candidates immediately; bounds on the maximum digit product and sum eventually dominate positional growth, limiting the number of digits that must be checked. Fixed points, transients, and longer cycles form branches of the same arithmetic dynamical system. This compression turns an apparently unbounded classification into a finite base-specific computation and makes clear which results depend on numeral representation rather than on representation-independent properties of the integer.

Abstract Reasoning¶

Representation move. From a number's digits in a fixed base, compute both digit sum and digit product and test the defining equality with the number. Constraint move. Use digit bounds, zero behavior, and place-value growth to restrict possible digit lengths and compositions. Construction move. Search by digit multisets or recurrences rather than blindly enumerating all integers, while restoring positional distinctions where they matter. Base-change move. Re-evaluate the property after changing radix because both operations are representation-dependent. Boundary move. A coincidental equality in one base does not make the property intrinsic to the integer, and sum-product numbers are not additive or multiplicative number classes in the usual sense.

Knowledge Transfer¶

Within the home domain. Sum-product numbers transfer across recreational mathematics, digit-function dynamics, and radix-dependent enumeration when an integer equals the sum plus product of its digits under a specified base and convention. Digit length, zeros, place value, and search bounds retain formal force. Beyond the home domain (C — formal class). The definition applies literally to any radix, but its members generally change with representation. Its boundary is semantic: the term does not describe a number closed under sum and product, an intrinsic algebraic property, or a model of combined processes. Numerical pattern alone carries no cross-domain mechanism.

Examples¶

Canonical¶

In decimal, 144 has digit sum 1+4+4=9 and digit product 1×4×4=16. Their product is 144, equal to the original integer, so 144 is a sum-product number. Any positive decimal numeral containing a zero has digit product zero and therefore cannot satisfy the equation. The equality also implies divisibility by the digit sum, making every positive fixed point a Harshad number, although most Harshad numbers are not sum-product numbers. Writing 144 in another base changes the digits and can change membership.

Mapped back: Decimal is the numeral base, 144 the represented integer, 9 the digit-sum statistic, and 16 the digit-product statistic. Their multiplication is the sum–product map satisfying the fixed-point equation and the Harshad consequence; another base tests the base-dependence rule.

Applied / In Practice¶

A search program enumerates fixed points in base b. It rejects zero-containing positive candidates early, computes a growth bound showing positional values eventually exceed every possible sum–product output for the same digit length, and checks only the resulting finite region. Iterating the map beyond fixed points also reveals transient trajectories and periodic cycles; these are recorded separately rather than called sum-product numbers. Unit tests convert digits directly in each base so decimal strings never leak into the calculation.

Mapped back: Early rejection uses the digit-product statistic, the growth proof the finite-state bound, and enumeration solves the fixed-point equation. Iterated transients and cycles are the dynamical extension, while explicit conversion enforces the base-dependence rule.

Structural Tensions¶

T1 — Identity versus admissible variation. Sum-product number must remain recognizable across legitimate variants. Admissible variation is bounded by this condition: Candidates are generated and checked exactly under one base and numeral convention. The stable element is expressed by this invariant: Sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory. Treating every surface change as a new abstraction fragments the identity, while allowing a change to the constitutive relation produces a false positive.

Diagnostic: After the proposed variation, can an analyst still establish this invariant: Sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory?

T2 — Recognition versus proxy. The domain needs observable or inferential evidence for Sum-product number, but the evidence is not automatically the identity. The working recognition rule is: the dynamical extension — transients and longer periodic cycles surrounding the period-one sum-product numbers. A familiar indicator can occur without the defining relation, and the relation can persist when a customary detector is unavailable.

Diagnostic: Does the evidence establish the defining claim—Sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory—or only a correlated sign?

T3 — Definition versus operational judgment. A compact definition aids reuse, whereas actual classification in recreational number theory can require expert decisions about boundary conditions, measurements, conventions, or exceptions. Viewing Fb(n)=Sb(n)Pb(n) as a map on the natural numbers places the concept in arithmetic dynamics. The definition must constrain those judgments without pretending that every admissible case can be recognized from a label alone.

Diagnostic: Which observation would make a competent practitioner reject the classification under the stated definition?

T4 — Scope versus overextension. Sum-product number has a genuine habitat in which candidates are generated and checked exactly under one base and numeral convention. Yet This is not primality or another intrinsic number class, zeros prevent positive fixed points through a zero product, leading zeros cannot be admitted, and computation without a proved bound does not establish completeness. A useful application map therefore has to be broad enough to cover recurring practice and narrow enough to exclude merely topical or metaphorical occurrences.

Diagnostic: Can the claimed application fill the same carrier and relation roles, or has only the name traveled?

T5 — Transfer versus domain accent. Knowledge about Sum-product number can travel within its home domain, and some structural lessons may travel farther. Sum-product numbers transfer across recreational mathematics, digit-function dynamics, and radix-dependent enumeration when an integer equals the sum plus product of its digits under a specified base and convention. What transfers must be separated from the specialist vocabulary, warrant, and closure conditions that remain anchored in recreational number theory.

Diagnostic: Is the receiving case a literal instance of Sum-product number, a co-instance of Classification, or only an analogy?

T6 — Autonomy versus reduction. Sum-product number is a strict specialization of Classification, but the edge does not erase the domain differentia. The broader node supplies only the necessary structural relation; recreational number theory supplies the carrier, warrant, boundary, and exception conditions expressed by this identity: Sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory. The entry is over-split if those conditions add no discriminating work and under-specified if the parent alone is used for cases that require them.

Diagnostic: Can a domain expert use the added conditions to distinguish Sum-product number from another case that equally instantiates Classification?

Structural–Framed Character¶

Sum-product number is mixed: structurally specifiable but materially dependent on its disciplinary frame. Its structural side consists of the carrier the numeral base — base $b$ fixing the digit alphabet and positional representation and the constitutive relation Sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory. Its framed side comes from recreational number theory, which fixes what the terms denote, what counts as evidence, and when a qualification or exception defeats the classification.

Across the principal tests, the entry is not merely a free-floating pattern. Evaluative weight: the identity can be stated descriptively even when its use has practical or normative consequences. Practice dependence: the dynamical extension — transients and longer periodic cycles surrounding the period-one sum-product numbers. Institutional stabilization: disciplinary conventions may stabilize the name and test without necessarily creating every underlying event or relation. Vocabulary portability: the invariant is Sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory. Import versus recognition: an outside case qualifies literally only if the same typed roles and collapse condition are available; otherwise the comparison is analogical.

The reusable remainder is Classification under a reviewed subsumption relation. That node preserves the necessary cross-domain organization after the recreational number theory-specific carrier, evidence, and exceptions are removed. Sum-product number remains autonomous because its recognition and collapse conditions distinguish cases that the parent alone leaves together.

Structural Core vs. Domain Accent¶

What is skeletal. The portable skeleton is a typed carrier organized by a constitutive relation, an invariant, a recognition test, and a collapse condition. Here the carrier is the numeral base — base $b$ fixing the digit alphabet and positional representation. The decisive relation is Sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory, which also states the controlling invariant at this level. Stripped of specialist nouns, this organization is represented by Classification.

What is domain-bound. recreational number theory supplies the actual objects or agents, admissible transformations, units or conventions, standards of warrant, and named exceptions. In this case, recognition requires evidence for the dynamical extension — transients and longer periodic cycles surrounding the period-one sum-product numbers. Admissible variation is bounded by the condition that candidates are generated and checked exactly under one base and numeral convention, and the classification collapses when membership depends on the numeral base and its digits rather than on arithmetic structure alone. These are constitutive differentia, not illustrative decoration.

Why it remains a domain-specific node. The reviewed DAG relation is subsumption to Classification. Outside recreational number theory, the parent captures only the reusable structural remainder. The specialist name remains literal only where the dynamical extension — transients and longer periodic cycles surrounding the period-one sum-product numbers can be established under the domain's standards of warrant.

This entry is a kind of Classification.

Immediate parent — Classification (subsumption). Sum-product number is a domain-specific kind of Classification: Sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory. The parent supplies the necessary broader identity—Sorting entities into discrete categories by explicit rules, turning unbounded variation into a finite, reusable map for downstream reasoning and action.—while the candidate adds the source-domain carrier, recognition rule, and failure conditions. The defining source account begins: A sum-product number in base b is a natural number equal to the product of the sum of its base-b digits and the product of those digits.
Nearest catalog surface declined — Square number. Its rematch score was 0.24045. Retrieval proximity did not establish synonymy or parentage; the carrier, invariant, and collapse condition remain different.
Related reasoning operations. Evidence, comparison, boundary testing, and representation can support a case without becoming additional DAG parents.

Relationships to Other Abstractions¶

Current abstraction Sum-product number Domain-specific

Parents (1) — more general patterns this builds on

Sum-product number is a kind of Classification Prime

Sum-product number is a domain-specific kind of Classification: Sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory.

Hierarchy path (1) — routes to 1 parentless root

Sum-product number → Classification

Neighborhood in Abstraction Space¶

Sum-product number sits in a moderately populated region (48^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Numeral Systems & Digit Algorithms (11 abstractions)

Nearest neighbors

Perfect digit-to-digit invariant — 0.94
Multiplicative Digital Root — 0.88
Cyclic Number — 0.86
Average Order of an Arithmetic Function — 0.85
Signedness — 0.85

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Classification. This is the reviewed immediate parent or structural prerequisite, not a synonym. Tell: retain Sum-product number only when the domain-specific relation Sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory. and its source-domain warrant are established; otherwise route the case to Classification.
Normal Number. This is the closest catalog retrieval surface, not an accepted synonym or parent. Tell: Ask which entry's carrier, invariant, and collapse test the case actually satisfies; shared vocabulary or a score of 0.75921 is insufficient.
Not an intrinsic magnitude class like primality. Membership depends on the numeral base and its digits rather than on arithmetic structure alone. Tell: Require the positive recognition condition that the dynamical extension — transients and longer periodic cycles surrounding the period-one sum-product numbers.
Not any number divisible by its digit sum. Every sum-product fixed point is Harshad, but divisibility alone does not force equality with sum times product. Tell: Replace the familiar surface feature and test whether sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory.
A detector, representation, or consequence. A method may reveal Sum-product number, a notation may describe it, and an outcome may follow from it without any of those being identical to the abstraction. Tell: Would the defining relation remain if the present detector, notation, or downstream effect changed?
A metaphorical transfer. A case outside the home domain may resemble the structure while lacking its native role types and standards of warrant. Tell: If only the general organization survives, route the comparison to Classification rather than treating it as another Sum-product number instance.

References¶

Frozen Wikipedia revision: https://en.wikipedia.org/wiki/Sum-product_number (revision 1356101655).
Wolfram MathWorld, ‘Sum-Product Number’: https://mathworld.wolfram.com/Sum-ProductNumber.html
OEIS A038369, numbers equal to the product of their digit sum and digit product: https://oeis.org/A038369
PlanetMath, ‘sum-product number’: https://planetmath.org/sumproductnumber The frozen Wikipedia revision is discovery provenance. The added sources are reference-grade authorities for the definition, formal relation, or professional practice summarized above; downstream historical or application claims remain bounded by the wording and scope of the cited source.

The frozen Wikipedia revision is discovery provenance. The cited source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; URL transport failure alone was not treated as substantive contradiction.